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In this paper, we consider a variational restoration model consisting of the I-divergence as data fitting term and the total variation semi-norm or nonlocal means as regularizer for removing multiplicative Gamma noise. Although the I-divergence is the typical data fitting term when dealing with Poisson noise we substantiate why it is also appropriate for cleaning Gamma noise. We propose to compute the minimizers of our restoration functionals by applying Douglas-Rachford splitting techniques, resp. alternating direction methods of multipliers. For a particular splitting, we present a semi-implicit scheme to solve the involved nonlinear systems of equations and prove its Q-linear convergence. Finally, we demonstrate the performance of our methods by numerical examples.

As first demonstrated by Chambolle and Lions the staircasing effect of the RudinOsher-Fatemi model can be reduced by using infimal convolutions of functionals containing higher order derivatives. In this paper, we examine a modification of such infimal convolutions in a general discrete setting. For the special case of finite difference matrices, we show the relation of our approach to the continuous total generalized variation approach recently developed by Bredies, Kunisch and Pock. We present splitting methods to compute the minimizers of the ℓ 2 2 -(modified) infimal convolution functionals which are superior to previously applied second order cone programming methods. Moreover, we illustrate the differences between the ordinary and the modified infimal convolution approach by numerical examples.

In this papers we analyze the minimization of seminorms L · on R n under the constraint of a bounded I-divergence D(b, H·) for rather general linear operators H and L. The I-divergence is also known as Kullback-Leibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data. Often H represents, e.g., a linear blur operator and L is some discrete derivative or frame analysis operator. We prove relations between the the parameters of I-divergence constrained and penalized problems without assuming the uniqueness of their minimizers. To solve the I-divergence constrained problem we apply first-order primal-dual algorithms which reduce the problem to the solution of certain proximal minimization problems in each iteration step. One of these proximation problems is an I-divergence constrained least squares problem which can be solved based on Morosov's discrepancy principle by a Newton method. Interestingly, the algorithm produces not only a sequence of vectors which converges to a minimizer of the constrained problem but also a sequence of parameters which convergences to a regularization parameter so that the corresponding penalized problem has the same solution as our constrained one. We demonstrate the performance of various algorithms for different image restoration tasks both for images corrupted by Poisson noise and multiplicative Gamma noise.

Motivated by a recent halftoning method which is based on electrostatic principles, we analyze a halftoning framework where one minimizes a functional consisting of the difference of two convex functions (DC). One of them describes attracting forces caused by the image gray values, the other one enforces repulsion between points. In one dimension, the minimizers of our functional can be computed analytically and have the following desired properties: the points are pairwise distinct, lie within the image frame and can be placed at grid points. In the two-dimensional setting, we prove some useful properties of our functional like its coercivity and propose to compute a minimizer by a forwardbackward splitting algorithm. We suggest to compute the special sums occurring in each iteration step by a fast summation technique based on the fast Fourier transform at non-equispaced knots which requires only O(m log m) arithmetic operations for m points. Finally, we present numerical results showing the excellent performance of our dithering method.